Introduction to Pump Rotordynamics - CiteSeerX

Introduction to Pump Rotordynamics Luis San Andrés Mast-Childs Tribology Professor Turbomachinery Laboratory Texas A&M University College Station, TX 77843-3123 USA [email protected]

ABSTRACT The lecture introduces the basic problems in the rotordynamics of turbomachinery, excessive vibration and instability. The acceptable performance of a turbomachine depends on the adequate design and operation of the bearing and seal elements supporting a rotor. Descriptions of the basic principles of lubrication follow with details on the operation of hydrodynamic and hydrostatic lubricated bearings and seals. The differences among these elements are highlighted with a brief account on their effects on rotordynamics. The basic equations for the modeling of linear rotor-bearing systems are given along with an example for the rotordynamics of a multiple stage compressor. Pump rotordynamics is introduced noting the major difference with other rotating systems, i.e. hydraulic side loads, static and dynamic, due to pressure changes in the volute and flow conditions in an impeller, dynamic forces from seals – neck ring and interstage and balance pistons, and impeller-rotor interaction forces. Accounting for the action of these elements is of importance to adequately predict the performance and troubleshoot the rotordynamics of high performance pumps. An example of rotordynamic analysis of a multiple-stage liquid pump stresses the differences between “wet’ and “dry” predictions, i.e. operation with and without the pumping liquid.

1.0 INTRODUCTION A turbomachinery is a rotating structure where the load or the driver handles a process fluid from which power is extracted or delivered to. Examples of turbomachines include pumps and compressors, gas and steam turbines, turbo generators and turbo expanders, turbochargers, APU (auxiliary power units), etc. Most turbomachinery is supported on oil lubricated fluid film bearings, although modern advances and environmental restrictions are pushing towards the implementation of process fluid bearings and even gas bearing applications. Fluid film bearings are used due to their adequate load support, good damping characteristics and absence of wear if properly designed and operated. Turbomachines also include a number of other mechanical elements which provide stiffness and damping characteristics and affect the dynamics of the rotor-bearing system. Impeller seals, floating ring seals, thrust collars and balance pistons are a few of these elements. The adequate operation of a turbomachine is defined by its ability to tolerate normal (and even abnormal) vibrations levels without affecting significantly its overall performance (reliability and efficiency). The rotordynamics of turbomachinery encompasses the structural analysis of rotors (shafts and disks) and the design of fluid film bearings and seals that determine the best dynamic performance given the required San Andrés, L. (2006) Introduction to Pump Rotordynamics. In Design and Analysis of High Speed Pumps (pp. 9-1 – 9-26). Educational Notes RTO-EN-AVT-143, Paper 9. Neuilly-sur-Seine, France: RTO. Available from: http://www.rto.nato.int/abstracts.asp.

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9-1

Introduction to Pump Rotordynamics operating conditions. This best performance is denoted by well-characterized natural frequencies (and critical speeds) with amplitudes of synchronous dynamic response within required standards and demonstrated absence of subsynchronous vibration instabilities. A rotordynamic analysis considers the interaction between the elastic and inertia properties of the rotor and the mechanical impedances from the fluid film bearing supports, oil seal rings, seals, etc. The most commonly recurring problems in rotordynamics are [1] 1. 2.

Excessive steady state synchronous vibration levels. Sub harmonic rotor instabilities.

Steady state vibration levels may be reduced by: a) Improving balancing. b) Modifying rotor-bearing systems: tune system critical speeds out of RPM operating range. c) Introducing damping to limit peak amplitudes at critical speeds that must be traversed. Sub harmonic rotor instabilities may be avoided by: a) Raising the natural frequency of rotor system as much as possible. b) Eliminating the instability mechanism, i.e. change bearing design if oil whip is present. c) Introducing damping to raise onset speed above the operating speed range. Rotordynamic instabilities have become more common as the speed and power of turbomachinery increased. These instabilities can sometimes be erratic, seemingly increasing vibration amplitudes for no apparent reason. A common denominator among many stability problems is that they tend to grow with time as the affected component(s) begins to wear or fatigue. For example, two typical destabilizing forces well documented in the technical literature are due to the aerodynamic effects of labyrinth seals and the hydrodynamic effects of lubricated cylindrical bearings and floating oil ring seals in centrifugal compressors. Load, gas molecular weight, and oil pressure and temperature are factors to generate severe problems in problematic turbomachinery. The detailed study of rotordynamics demands accurate knowledge of the particular mechanical elements supporting the rotor, i.e. fluid film bearings and seals.

2.0 DESCRIPTIONS OF FLUID FILM BEARINGS Fluid film bearings are machine elements designed to produce smooth (low friction) motion between solid surfaces in relative motion and to generate a load support for mechanical components. The lubricant or fluid between the surfaces may be a liquid, a gas or even a solid (coating). Fluid film bearings, if well designed, are able to support static and dynamic loads, and consequently, their effects on the performance of rotating machinery are of great importance. These notes focus on the analysis of bearings with a full film separating the mechanical surfaces. The word film implies that the fluid thickness (gap or clearance) separating the surfaces is several orders of magnitude smaller than the other dimensions of the bearing, i.e. width and length. Yet the film thickness is larger than the micro asperities in the surfaces thus warranting operation without contact of surfaces. The basic operational principles of fluid film bearings are hydrodynamic, hydrostatic or hybrid (a combination of the former two).

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Introduction to Pump Rotordynamics

2.1

Hydrodynamic or Self-Acting Fluid Film Bearings

In hydrodynamic bearings there is relative motion between two mechanical surfaces with a particular “wedge like” shape, See Figure 1. The fluid dragged into the film generates a hydrodynamic pressure field able to support an externally applied load, static and/or dynamic. In general [2]: Advantages

Disadvantages

Do not require external source of pressure. Thermal effects affect performance if film thickness is Fluid flow is dragged into the convergent too small or available flow rate is too low. gap in the direction of the surface relative Require of surface relative motion to generate load motion. support. Support heavy loads. The load support is a function of the lubricant viscosity, surface Induce large drag torque (power losses) and potential speed, surface area, film thickness and surface damage at start-up (before lift-off) and touch down. geometry of the bearing. Long life (infinite in theory) without wear of Potential to induce hydrodynamic instability, i.e. loss of effective damping for operation well above critical surfaces. speed of rotor-bearing system. Provide stiffness and damping coefficients of large magnitude.

2.2

Hydrostatic or Externally-Pressurized Fluid Film Bearings

In hydrostatic bearings, an external source of pressurized fluid forces the lubricant to flow between the surfaces, thus enabling their separation and the ability to support a load without surface contact, and most importantly, without relative motion. See Figure 2 for a typical geometry. In general [2]: Advantages

Disadvantages

Support very large loads. The load support is Require ancillary equipment. Larger installation and a function of the pressure drop across the maintenance costs. bearing and the area of fluid pressure action. Need of fluid filtration equipment. Loss of performance Load does not depend on film thickness or with fluid contamination. lubricant viscosity. High power consumption because of pumping losses. Long life (infinite in theory) without wear of Potential to induce hydrodynamic instability in hybrid surfaces mode operation. Provide stiffness and damping coefficients of very large magnitude. Excellent for exact Potential to show pneumatic hammer instability for highly compressible fluids, i.e. loss of damping at low positioning and control. and high frequencies of operation due to compliance and time lag of trapped fluid volumes.

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Introduction to Pump Rotordynamics Journal rotation Pressure

Relative motion

Pressure

Hydrodynamic wedge

Fluid

Plain journal bearing Slider bearing

Figure 1: Schematic Views of Hydrodynamic (Self-Acting) Fluid Film Bearing.

Pressure

Ps Pr

restrictor

Fluid at Ps

recess film

Hydrostatic journal bearing

Figure 2: Schematic Views of Hydrostatic (Pressurized) Fluid Film Bearing.

Figure 3 depicts schematic views of hydrodynamic journal bearings, single and multiple lobes, and multiple pads. The cylindrical bearings feature feed ports for lubricant inlet as well as pad or lobes machined with a preload to generate a film wedge upon shaft rotation.

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PLAIN JOURNAL BEARING

PARTIAL ARC JOURNAL BEARING

FLOATING RING JOURNAL BEARING

Dam Top half

Bottom half

PRESSURE DAM JOURNAL BEARING

Relief Groove

TILTING PAD JOURNAL BEARING

Figure 3: Schematic Views of Typical Cylindrical Journal Bearings.

2.3

Squeeze Film Dampers

Normal motions can also generate hydrodynamic pressures in the thin film separating two surfaces. This squeeze film action mechanism works effectively only for compressive loads, i.e. those forcing the approach of one surface to the other. Squeeze film dampers are routinely implemented to reduce RTO-EN-AVT-143

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Introduction to Pump Rotordynamics vibration amplitudes and isolate structural components in gas jet engines, high performance compressors, and occasionally in water pumps. A squeeze film damper consists of an inner non rotating journal and a stationary outer bearing, both of nearly identical diameters. Figure 4 shows an idealized schematic of this type of fluid film bearing. A journal is mounted on the external race of a rolling element bearing and prevented from spinning with loose pins or a squirrel cage that provides a centering elastic mechanism. The annular thin film, typically less than 0.250 mm, between the journal and housing is filled with a lubricant provided as a splash from the rolling bearing elements lubrication system or by a dedicated pressurized delivery. In operation, as the journal moves due to dynamic forces acting on the system, the fluid is displaced to accommodate these motions. As a result, hydrodynamic squeeze film pressures exert reaction forces on the journal and provide for a mechanism to attenuate transmitted forces and to reduce the rotor amplitude of motion [3]. anti-rotation pin shaft journal ball bearing

lubricant film housing

Figure 4: Typical Squeeze Film Damper Configuration.

2.4

Annular Pressure Seals

Radial seals (annular, labyrinth or honeycomb) separate regions of high pressure and low pressure in rotating machinery and their function is to minimize the leakage and improve the overall efficiency of a rotating machine extracting or delivering power to a fluid. Typical applications include neck ring seals on impeller eyes and inter stage seals as well as balance pistons in pump and compressor applications, as depicted in Figure 5. Seals have larger clearances than load carrying bearings. Yet their impact on the rotordynamics of turbomachinery is of importance since seals are located at rotor positions where large vibrations occur [4].

Interstage Seal

Impeller Eye Seal

Balance Piston Seal

Figure 5: Seals in a Multistage Centrifugal Pump or Compressor. 9-6

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Introduction to Pump Rotordynamics Extensive testing has shown that seals with macroscopic roughness; i.e. textured stator surfaces, offer major improvements in reducing leakage as well as cross-coupled stiffness coefficients [5]. Figure 6 depicts two textured seals and a conventional labyrinth seal (teeth on stator). A textured surface like a round-hole pattern or a honeycomb increases the friction thus reducing leakage, and aids to retard the development of the circumferential flow velocity -the physical condition generating the cross-coupled stiffness coefficients. In the past 10 years, compressor and pump manufacturers, as well as users, are implementing textured seals with great commercial success [6]. Hole-Pattern Seal

Unwrap

Honeycomb Seal

Unwrap

Labyrinth Seal

Figure 6: Hole-Pattern, Honeycomb and Labyrinth Seal Configurations.

3.0 BASICS OF A ROTORDYNAMIC ANALYSIS The objectives of a rotordynamic analysis are: a) To model the rotor (shaft and disks) and to determine its free-free natural frequencies. b) To model the fluid film bearing and seal elements and to calculate the mechanical impedances (stiffness, damping and inertia force coefficients connecting the rotor to its casing). c)

To perform an eigenvalue analysis, i.e. to predict the damped natural frequencies and damping ratios for the different modes (rigid and elastic) of vibration of the rotor as the rotor speed increases to values well above its design operating conditions. Positive damping ratios evidence the absence of rotordynamic instability.

d) To perform a synchronous response analysis to calibrated imbalances in order to predict the maximum amplitudes of vibration, the safe passage through critical speeds and to estimate the loads transmitted through the bearing supports. e) To certify the reliable performance of the rotor-bearing system satisfying established engineering criteria (API qualification) and to emit recommendations to improve the system performance (response and stability).

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Introduction to Pump Rotordynamics It is important to stress that the tasks (objectives) above need of extensive experimental and field support verification. Analysis without adequate field or shop measurements is usually not very useful in rotordynamics.

3.1

Equations of Motion for Rotor-Bearing System

The general equations for lateral motions of a rotor are [4]:

([M ] + [N ])R {u}− Ω [G ]R {u}+ [K ]R {u}= {F (u, u, t )}

(1)

where [M] and [N] are global translational and rotary mass matrices, [G] and [K] are gyroscopic and stiffness matrices, Ω is the rotor speed, {u}represent the rotor displacements (translations and rotations), and {F (u , u , t )} denote bearing and seal reaction forces and the distributed imbalance vector, for example, The in-house finite element rotordynamics code[7] is based on Timoshenko beam theory [8] and implements component-mode synthesis [9] to model multiple shaft rotors. The program interfaces with a number of bearing and seal codes also developed in-house.

3.2

Representations of Bearing and Seal Forces

In a linear lateral rotordynamics model, bearing and seal reaction forces are linear functions of the rotor motion. Bearing dynamic forces along directions (X, Y) perpendicular to the rotor spin axis (Z) are represented in terms of force coefficients (See Figure 7), i.e.

 FX   K XX F  =− K  Y  YX

K XY   X  C XX  − K YY  B  Y   CYX

C XY   X    CYY  B  Y 

(2)

where [K]B and [C]B denote matrices of stiffness and damping force coefficients. The force coefficients are strictly valid for small amplitude motions or perturbations about an equilibrium condition. The static load acting on each bearing and shaft speed determine the equilibrium state. Force coefficients for oillubricated bearings are frequency independent parameters, though changing with the operating shaft speed and load condition. Fluid film bearings handling compressible liquids and gases generate frequency dependent force coefficients.

Y

δY

Z

δX

X Figure 7: DOF Lateral Displacements (X,Y) and Angulations (δX, δY).

The representation of liquid seal forces for lateral motions (X, Y) is given as [4]

 FX   K XX F  =− K  Y  YX 9-8

K XY   X  C XX  − K YY  S  Y   CYX

C XY  CYY  S

 X   M XX   −  Y   M YX

M XY  M YY  S

 X    Y 

(3)

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Introduction to Pump Rotordynamics where [K]S, [C]S and [M]S represent matrices of stiffness, damping and inertia force coefficients. Added mass coefficients are of importance in liquid seals due to the large density of the fluid pumped. Dynamic reaction forces in gas seals are nowadays represented as

 FX   K XX (ω ) K XY (ω )  X  C XX (ω ) C XY (ω )  F  = −  K (ω ) K (ω )   Y  −  C (ω ) C (ω )  YY YY  Y  YX  S    YX S

 X    Y 

(4)

with force coefficients showing a complicated dependency on frequency [5]. Balance pistons are represented as long seals, and thus these elements can also generate bending moments (MX,Y) due to lateral displacements, as well as reaction forces due to dynamic angulations (φX, φY) around the (X, Y) axes, i.e. [4]  FX   FY  MX   MY

 K XX     K YX =-    K δXX     - K δXY

K XY

K XδX

K XX

K XδY

K δXY

K δXδX

K δXX

K δXδY

K XδY   X   C XX C XY   K XδX   Y   C YX C XX   -  K δXδY  φ X   C δXX C δXY     φ K δXδX   Y   C δXY C δXX

C XδX C XδY C δXδX C δXδY

C XδY   X   M XX    C XδX   Y   M YX  - C δXδY  φ X   M δXX    C δXδX  φY   M δXY

M XY

M XδX

M XX

M XδY

M δXY

M δXδX

M δXX

M δXδY

M XδY   X    M XδX   Y     M δXδY  φ X     M δXδX  φY 

(5)

Incidentally, aerodynamic (or hydraulic) forces from impeller interactions in centrifugal compressors (or pumps) and axial flow turbines are also represented in a similar form. Forces and moments from support bearings and seal force coefficients are integrated into the equations of motion (1) rendering

where

[ M ] {u}− [[C ] − Ω [G ]R ]{u}+ [K ]{u}= {Fext (u, u , t )}

(6)

[M ]= ([ M ] + [ N ])R ∪ ∑ [ M ] S ; [K ]= [K ]R ∪ ∑ [ K ] B ∪ ∑ [ K ] S ; [C ]= ∪ ∑ [C ] B ∪ ∑ [C ] S ;

(7)

are the system inertia, stiffness and damping matrices. Above sub-indices B and S represent the contributions of bearing and seals. To determine the damped natural frequencies of the rotor-bearing system, the homogeneous form of Eqn. (6) is solved. The general solution is of the form, {u} = {v} est which renders the following eigenvalue problem.

[[K ]+ s [C ]− Ω s [G ]

R

]

+ s 2 [M ] {v}= {0}

(8)

Solution of the characteristic equation (8), a polynomial in s, leads to the set of eigenvalues s=λ+ i ω and eigenvectors {v}, also known as natural damped mode shapes. Note that the system eigenvalues are rotor speed dependent (Ω). For stability, the real part of all eigenvalues must be negative, i.e. λ 1.

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Introduction to Pump Rotordynamics •

The shroud generates most of the destabilizing forces in a pump impeller [16].

•

Enlarged clearance in neck ring seals aggravate the generation of cross-coupled forces since the swirl flow into the seal increases. Enlarged clearances are the result of transient rub events in operation of a pump.

•

At reduced flow rates (< QBEP), the impeller cross-coupled stiffnesses are lower.

•

Impeller-volute interaction forces are small and benign [16].

•

In a radial flow impeller cross-coupled stiffnesses are small, i.e. the destabilizing force is negligible since the projected axial area of the shroud is quite small.

Childs has advanced two bulk-flow analyses [21, 22] for the prediction of impeller-interaction forces in pumps and compressors, respectively. The first analysis considers the secondary flow through a pump impeller shroud and derives dynamic force coefficients as a function of the operating condition and whirl frequency ratio. Comparison of predictions to Sulzer’s test data is good for the whirl frequency ratio and reasonable for cross-coupled stiffness and direct damping. Predictions of other coefficients are rather poor. Note that the bulk-flow model, by integrating the velocity film across the thin film clearance, cannot reproduce regions of flow recirculation. CFD studies demonstrate these flow zones extend over the entire secondary flow path, in particular when tight exit seals are in place. Childs concludes that leakage flowing radially inward towards the neck ring seals generates a large inlet swirl, i.e. mean circumferential flow larger than 50% of rotor surface speed, which generates large crosscoupled stiffnesses able to induce rotordynamic instability, i.e. reduce stability margin. Childs [16] states that the impeller-rotor interaction forces increase sharply with decreasing clearance between the impeller shroud and the pump housing. Ehrich [15] states that enlarged clearances in the exit seal due to wear or damage will increase the circumferential flow velocity over the shroud and into the seal, thus enlarging the cross-coupled forces [15]. In the most recent analysis [21], Childs performs a complete bulk-gas flow analysis integrating the secondary flows through the front shroud and its neck-ring seal, as well as the leakage (inward or outward) flow through the impeller back face and the inter stage seal. The predictions, applicable to pumps, are in agreement with test data for moment coefficients when using unrealistic values for the inlet losses into the eye lacking labyrinth seals. Other predictions for a compressor stage are reasonable; albeit showing “fluidic” resonances at low whirl frequency ratios. In [21], individual forces from each element are calculated. The authors find that the eye seal and interstage seals account for 80% and 18% of the crosscoupled stiffness, while the front shroud contributes with just 2.2%. The effect of the back shroud is entirely negligible, even for radial inflow. Thus, the impeller-rotor interaction forces in compressors appear to be mostly affected by the seals. Industry uses the API standard Wachel-formula [16] to predict the aerodynamic cross-coupled force in a compressor. The results in [21] show more conservative results, i.e. less destabilizing forces, than the accepted formula. However, there have been instances when compressors have become unstable in spite of their design with consideration of Wachel’s aerodynamic force. Childs [21] concludes that the deficiency in predictive analysis properly validated by experimental results remains. Impeller induced forces are thought as the “last” great unknown in modern rotordynamics.

4.3

Example of Pump Rotordynamics Analysis

A linear rotordynamics analysis for an 11-stage pump follows. Figure 17 depicts the rotor structural model. The pump rotor weighs 363 lb and its length is 97.45 inch. The shaft mean diameter at the impeller locations is 2.5 inch in and the bearing span is 79.4 inch. The impellers are 8.12 inch in diameter (Do) with a throat discharge width (b2) of 0.36 inch. The liquid density is 1.5 slug/ft3 (775 kg/m3). 9 - 20

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11-Stage Liquid Pump Analysis Support JBs (2), Stage Seals (11), Stage Impeller Coefficients, Sta. 38 End Seal w/Moment

30

Shaft Radius, inches

20

Tilting 4 Pad bearing D=2.5 in, L=1 in c=3 mil, r=0.25, of=0.55

11 Impellers D=8.2 in, b=0.36 in

10 Shaft1 1

5

15

10

20

25

30

35

40

45

0

neck ring seal

-10

50

Shaft1 55

balance piston

-20

-30 0

20

40

60

80

100

Axial Location, inches

Figure 17: Rotordynamics Model of 11-Stage Pump.

Two identical four-tilting pad bearings support the pump. The bearing diameter and length equal 2.5 inch and 1.0 inch, respectively, and the radial clearance is 0.003 inch (76 micron). Each pad arc is 80° with pivot offset and assembly preload equal to 0.55 and 0.25, respectively. The lubricant viscosity and specific gravity are 2 micro-Reyns (13.8 cPoise) and 0.85. Each bearing supports half the rotor weight (Load between pad configuration). Bearing and impeller seal force coefficients are calculated with XLTRC2 software suite based on the analyses in [22] and [23]. Impeller force coefficients are estimated from tabular results based on the Sulzer experimental findings, see Table 6, ref.[16], for operation at Best Efficiency Point. In the rotor model, there are 2 bearing stations (6, 47); 12-seal stations (16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, end seal 38); and 11-impeller stations (17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37).

30

30

20

20 Shaft Radius, inches


The rotor is rather flexible. The first two free-free modes are at just 2,346 rpm and 5,469 rpm as shown in Figure 18.

10 0 -10 -20

10 0 -10 -20

-30

-30

0

20

40

60


2,346 rpm

80

100

0

20

40

60

80

100


5,469 rpm

Figure 18: Free-Free (Undamped) Modes of Multi-Stage Pump.

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Introduction to Pump Rotordynamics The eigenvalue analysis predicts the damped natural frequencies, damping ratios and critical speeds for a range of pump rotational speeds. Table 7 summarizes the predictions for the “dry” and “wet” conditions. The dry condition only accounts for the tilting pad bearing force coefficients. The wet case includes the effects of the bearings, seals and impellers on the pump rotordynamics. Table 7: Critical Speeds, Natural Frequencies and Damping Ratios for Multiple Stage Pump – DRY Condition

DRY Critical speeds

WET Critical speeds

mode

cpm 1340 5155

damp ratio

number

0.007 0.013

2 8

mode

cpm 400 1482 3574

damp ratio

number

-0.024 0.023 0.026

2 4 10

Forward modes damp ratio Speed cpm 3500 rpm 0.007 1340 0.013 5140

mode number 2 8

Forward modes damp ratio cpm Speed 3500 rpm -0.169 716 -0.010 1451 0.026 3574 0.010 11340

mode number 2 4 10 12

For the “dry” condition, the rotor will traverse two lightly damped critical speeds (1340 & 5155 rpm) in the speed range from 1000 to 6000 rpm. Figure 19 depicts the map of damped natural frequencies and damping ratios versus shaft speed for the forward whirl modes. The rotor modes at the design speed of 3,500 rpm are also depicted in the Figure.

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Natural Frequency, cpm

9000 8000

1X

2nd crit speed 5155 cpm

7000 6000

mode 4

5000 4000

first crit speed 1340 rpm

3000 2000

mode 2

1000 0 0.

1000.

2000.

3000.

4000.

5000.

6000.

7000.

Rotor Speed, rpm

0.050

0.04

0.040

Damping Ratio

Damping Ratio

Rotordynamic Root Locus Plot 0.05

0.03 0.02

mode 4

0.01

mode 2

0

0.030 0.020 mode 4

mode 2 0.010 0.000

0.

1000.

2000.

3000.

4000.

5000.

6000.

7000.

Rotor Speed, rpm

Mode 2

30

0

1000

2000

3000 4000 Natural Frequency, cpm

6000

7000

Damped Eigenvalue Mode Shape Plot

forward backward

20 Shaft Radius, inches

5000

f=1339.3 cpm d=.0068 zeta N=3500 rpm

10 0 -10 -20 -30 0

20

40

60

80

100


Mode 4 Damped Eigenvalue Mode Shape Plot

30

forward backward


20

f=5140.3 cpm d=.0127 zeta N=3500 rpm

10 0 -10 -20 -30 0

20

40

60

80

100


Figure 19: Natural Frequencies, Damping Ratios and Mode Shapes for Pump - DRY Case, Forward Whirl Damped Mode Shapes at Operating Speed (3,500 rpm).

Note that the rotor mode shapes are rather flexible with nodes at the bearing locations, hence the small damping ratios. Figure 20 shows the damped natural frequencies and mode shapes for the “wet” condition. Note that the critical speeds have “dropped” substantially due to the seals mainly. Furthermore, the damping ratios are negative for most of the operating speed range. The multiple stage pump will be unstable nearly from start up conditions.

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Natural Frequency, cpm

9000 8000 1X

7000 2nd crit speed 1482 rpm

6000 5000 4000

3rd crit speed 3574 rpm

first crit speed 400 rpm

mode 10

3000 mode 4

2000 1000 0 0.00

mode 2 1000.00

2000.00

3000.00

4000.00

5000.00

6000.00

7000.00

Rotor Speed, rpm

Rotordynamic Root Locus Plot mode 10

0.05

0.050 0.000

-0.05 mode 4

-0.1 -0.15

mode 12

mode 10

mode 12

Damping Ratio

Damping Ratio

0

mode 2

-0.2

-0.050

mode 4

-0.100 -0.150

mode 2

-0.200 -0.25 0.

1000.

2000.

3000.

4000.

5000.

6000.

7000.

-0.250 0.0

Rotor Speed, rpm

2000.0

4000.0 6000.0 8000.0 Natural Frequency, cpm

10000.0

12000.0

forward backward forward backward

f=716. cpm d=-.1695 zeta N=3500 rpm

f=3574. cpm d=.0255 zeta N=3500 rpm

forward backward f=1451.5 cpm d=-.0101 zeta N=3500 rpm

Mode 2

Mode 10 Mode 4

Figure 20: Natural Frequencies, Damping Ratios and Mode Shapes for Pump - WET Case, Forward Whirl Damped Mode Shapes at Operating Speed (3,500 rpm).

4.4

Closure

Synchronous response analysis is not conducted since the pump is unstable for nearly all shaft speeds. The rotordynamics predictions evidence the need to redesign the pump. The recommendations to follow are: •

Increase shaft diameter to make rotor more rigid thus raising (free-free mode) natural frequencies.

•

Decrease bearing span to allow for more shaft motions at the bearing locations (avoid location of bearings at nodal points).

•

Add inlet-anti swirl brakes into neck ring seals to reduce development of circumferential flow speed and ameliorate generation of cross-coupled destabilizing forces. Add shunt injection on balance piston.

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Introduction to Pump Rotordynamics •

Verify predictions for impeller-interaction forces. Use latest test data if available.

•

Re-design oil-lubricated bearings. Current configuration shows too large stiffness and too much damping. Enlarge slightly the clearances if applicable.

The following two lectures provide the fundamentals of fluid film bearing and seal design and their effect on the rotordynamics of turbomachinery. The importance of fluid inertia and flow turbulence on modern bearing and seal configurations is detailed along with applications to a cryogenic liquid turbopump.

REFERENCES [1]

Self-Excited Vibrations in High-Performance Turbomachinery, F. Ehrich & D. Childs, Mechanical Engineering, 106, 5, pp. 66-79, May 1984.

[2]

Introduction to Modern Lubrication, L. San Andrés, Lecture Notes (#1) in Modern Lubrication, http://phn.tamu.edu/TRIBGroup, 2002.

[3]

Design and Application of Squeeze Film Dampers in Rotating Machinery, F. Zeidan, L. San Andrés & J. Vance, Proc. of the 25th Turbomachinery Symposium, TAMU, pp. 169-188, 1996.

[4]

Turbomachinery Rotordynamics, (chapter 4), D. Childs, John Wiley & Sons, Inc., 1993.

[5]

Annular Gas Seals and Rotordynamics of Compressors and Turbines, D. Childs & J. Vance, Proc. Of the 26th Turbomachinery Symposium, TAMU, pp. 201-220, 1997.

[6]

Gas Damper Seal Test Results, Theoretical Correlation, and Applications in Design of High-Pressure Compressors, P. De Choudhury, F. Kushner & J. Li, Proc. Of the 29th Turbomachinery Symposium, TAMU, 2001.

[7]

XLTRC2 Rotordynamics Software Suite, Turbomachinery Laboratory, Texas A&M University, 2002.

[8]

A Finite Rotating Shaft Element Using Timoshenko Beam Theory, H.D. Nelson, ASME Journal of Mechanical Design, 102, pp. 793-803, 1980.

[9]

Transient Analysis of Rotor-Bearing System Using Component Mode Synthesis, H.D. Nelson & H. Meacham, ASME Paper 81-GT-10, Gas Turbine Conference.

[10] A Virtual Tool for Prediction of Turbocharger Nonlinear Dynamic Response: Validation Against Test Data, L. San Andrés, L., J.C. Rivadeneira, K. Gjika, C. Groves, and G. LaRue, ASME Paper GT 2006-90873, 2006. [11] Pump Rotordynamics made Simple, M. Carbo & S. Malanoski, Proc. Of the 15th International Pump Users Symposium, TAMU, pp. 167-203, 1998. [12] Hydraulic Institute Centrifugal Pump Design and Application, pp. 103-105, 1994. [13] Stepanoff, A. J. "Centrifugal and Axial Flow Pumps-Theory, Design, and Application," 2nd Edition, Wiley, New York, NY. 1957. [14] Rotating Machinery Vibration, (Chapter 6), M. Adams, M. Dekker, Inc., 2001. [15] Handbook of Rotordynamics, F. Ehrich, Mc-Graw-Hill Co., 1992. RTO-EN-AVT-143

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Introduction to Pump Rotordynamics [16] Turbomachinery Rotordynamics, (chapter 6), D. Childs, John Wiley & Sons, Inc., 1993. [17] Annular Pressure Seals and Hydrostatic Bearings, L. San Andrés, Design and Analysis of High Speed Pumps, Von Karman Institute for Fluid Dynamics, Lecture Series, March 2006. [18] Aerodynamically Induced Radial Forces in a Centrifugal Gas Compressor - Part 1: Experimental Measurement, J. Moore & M. Flathers, ASME Paper 96-GT-120, 1996. [19] Experimental Measurements of Hydrodynamic Stiffness Matrices for a Centrifugal Pump Impeller, D. Chamieh, A.J. Acosta, T.K. Caughey & R. Franz, Proc. Of the Workshop in Rotordynamics Instability Problems in High Performance Turbomachinery, TAMU, NASA CP-2250, pp. 382-398, 1982. [20] Measurement of Hydrodynamic Matrices of Boiler Feed Pump Impellers, U. Bolleter, A. Wyss, I. Welte & R. Struchler, ASME Journal of Vibrations, Acoustics, Stress reliability and Design, 109, 1987. [21] Rotordynamic Stability predictions for Centrifugal Compressors Using a Bulk-Flow Model to Predict Impeller Shroud Force and Moment Coefficients, M. Gupta & D. Childs, ASME Paper GT2006-90374, 2006. [22] Turbulent Flow, Flexure-Pivot Hybrid Bearings for Cryogenic Applications, L. San Andrés, ASME Journal of Tribology, Vol. 118, 1, pp. 190-200, 1996. [23] Effect of Shaft Misalignment on the Dynamic Force Response of Annular Pressure Seals, L. San Andrés, STLE Tribology Transactions, Vol. 36, 2, pp. 173-182, 1993.

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